A mediation is an interactive process in which two parties try to resolve a conflict. This method requires the help of a mediator, who is an impartial third party who is specialized in the resolution of conflict through methods such as communication and negotiation techniques.
On the other hand, a neutral case evaluation is a process in which a neutral person examines the evidence and gives his opinion on the case. This is a relatively quick and easy procedure that can help both parties manage their expectations.
I would recommend a mediation in this case. It is clear that both parties have great interest in the dog. However, it is also likely that both parties would want to benefit the dog more than themselves. Because of this, I believe that they would be more open to talking and discussing alternatives. Through mediation, it could be possible for both parties to be reasonably satisfied with the outcome.